Recent questions and answers in Engineering Mathematics

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1

GATE2019 CE-1: 1
Which one of the following is correct? $\lim_{x\rightarrow 0} ( \frac{sin4x}{sin2x})=2 $ and $\lim_{x\rightarrow 0} ( \frac{tanx}{x})=1$ $\lim_{x\rightarrow 0} ( \frac{sin4x}{sin2x})=1$ and $\lim_{x\rightarrow 0} ( \frac{tanx}{x})=1$ ... $\lim_{x\rightarrow 0} ( \frac{sin4x}{sin2x})=2$ and $\lim_{x\rightarrow 0} ( \frac{tanx}{x})= \infty$

answered 3 days ago in Calculus by Kushagra गुप्ता (140 points)

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2

GATE2017 CE-2-19
The divergence of the vector field $V=x^2 i + 2y^3 j + z^4 k$ at $x=1, \: y=2, \: z=3$ is ________

asked Aug 7, 2019 in Calculus by gatecse (3.9k points)

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3

GATE2017 CE-2-27
Consider the following definite integral: $I= \int_0^1 \frac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is $\frac{\pi ^3}{24}$ $\frac{\pi ^3}{12}$ $\frac{\pi ^3}{48}$ $\frac{\pi ^3}{64}$

asked Aug 7, 2019 in Calculus by gatecse (3.9k points)

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4

GATE2017 CE-2-28
If $A = \begin{bmatrix} 1 & 5 \\ 6 & 2 \end{bmatrix}$ and $B= \begin{bmatrix} 3 & 7 \\ 8 & 4 \end{bmatrix}, \: AB^T$ is equal to $\begin{bmatrix} 38 & 28 \\ 32 & 56 \end{bmatrix}$ $\begin{bmatrix} 3 & 40 \\ 42 & 8 \end{bmatrix}$ $\begin{bmatrix} 43 & 27 \\ 34 & 50 \end{bmatrix}$ $\begin{bmatrix} 38 & 32 \\ 28 & 56 \end{bmatrix}$

asked Aug 7, 2019 in Linear Algebra by gatecse (3.9k points)

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5

GATE2017 CE-2-29
Consider the following second-order differential equation: $y’’ – 4y’+3y =2t -3t^2$. The particular solution of the differential solution equation is $ – 2 -2t-t^2$ $ – 2t-t^2$ $2t-3t^2$ $ – 2 -2t-3 t^2$

asked Aug 7, 2019 in Partial Differential Equation (PDE): by gatecse (3.9k points)

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2 answers

6

GATE2019 CE-2: 43
A series of perpendicular offsets taken from a curved boundary wall to a straight survey line at an interval of $6 \: m$ are $1.22, \: 1.67, \: 2.04, \: 2.34, \: 2.14, \: 1.87$, and $1.15 \:m$. The area (in $m^2$, round off to 2 decimal places) bounded by the survey line, curved boundary wall, the first and the last offsets, determined using Simpson’s rule, is ________

answered Feb 20, 2019 in Numerical Methods by moinkhan (140 points)

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7

GATE2019 CE-1: 2
Consider a two-dimensional flow through isotropic soil along $x$ direction and $z$ direction. If $h$ is the hydraulic head, the Laplace's equation of continuity is expressed as $\frac{\partial h}{\partial x}+ \frac{\partial h}{\partial z} = 0$ ...

asked Feb 14, 2019 in Partial Differential Equation (PDE): by Arjun (2.8k points)

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8

GATE2019 CE-1: 4
For a small value of $h$, the Taylor series expansion for $f(x+h)$ is $f(x)+h{f}’ (x) + \frac{h^2}{2!}{f}’’(x) + \frac{h^3}{3!}{f}’'’(x)+...\infty $ $f(x)-h{f}’ (x) + \frac{h^2}{2!}{f}’’(x) - \frac{h^3}{3!}{f}’'’(x)+...\infty $ $f(x)+h{f}’ (x) + \frac{h^2}{2}{f}’’(x) + \frac{h^3}{3}{f}’'’(x)+...\infty $ $f(x)-h{f}’ (x) + \frac{h^2}{2}{f}’’(x) - \frac{h^3}{3}{f}’'’(x)+...\infty $

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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9

GATE2019 CE-1: 23
The probability that the annual maximum flood discharge will exceed $25000 \: m^3/s$, at least once in next $5$ years is found to be $0.25$. The return period of this flood event (in years, round off to $1$ decimal place is ________

asked Feb 14, 2019 in Probability and Statistics by Arjun (2.8k points)

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10

GATE2019 CE-1: 26
Which one of the following is NOT a correct statement? The function $\sqrt[x]{x}, \: (x>0)$, has the global maxima at $x=e$ The function $\sqrt[x]{x}, \: (x>0)$, has the global minima at $x=e$ The function $x^3$ has neither global minima nor global maxima The function $\mid x \mid$ has the global minima at $x=0$

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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11

GATE2019 CE-1: 44
Consider the ordinary differential equation $x^2 \dfrac{d^2y}{dx^2} – 2x \dfrac{dy}{dx} +2y=0$. Given the values of $y(1)=0$ and $y(2)=2$, the value of $y(3)$ (round off to $1$ decimal place), is _________

asked Feb 14, 2019 in Ordinary Differential Equation (ODE) by Arjun (2.8k points)

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12

GATE2019 CE-2: 3
The following inequality is true for all $x$ close to $0$. $2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$ What is the value of $\underset{x \to 0}{\lim} \dfrac{x \sin x}{1 – \cos x}$? $0$ $1/2$ $1$ $2$

asked Feb 12, 2019 in Calculus by Arjun (2.8k points)

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13

GATE2019 CE-2: 26
The probability density function of a continuous random variable distributed uniformly between $x$ and $y$ (for $y>x$) is $\dfrac{1}{x-y}$ $\dfrac{1}{y-x}$ $x-y$ $y-x$

asked Feb 12, 2019 in Probability and Statistics by Arjun (2.8k points)

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14

GATE2019 CE-2: 28
An ordinary differential equation is given below; $\left ( \dfrac{dy}{dx} \right ) (x \text{ ln } x)=y$ The solution for the above equation is (Note: $K$ denotes a constant in the options) $y=K x \text{ ln } x$ $y=K x e^x$ $y=K x e^{-x}$ $y=K \text{ ln } x$

asked Feb 12, 2019 in Ordinary Differential Equation (ODE) by Arjun (2.8k points)

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15

GATE2019 CE-2: 35
The inverse of the matrix $\begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}$ is $\begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix}$ ...

asked Feb 12, 2019 in Linear Algebra by Arjun (2.8k points)

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16

GATE2016-2-4
$X$ and $Y$ are two random independent events. It is known that $P(X)=0.40$ and $P(X \cup Y^C)=0.7$. Which one of the following is the value of $P(X \cup Y)$? $0.7$ $0.5$ $0.4$ $0.3$

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (11.8k points)

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17

GATE2016-2-5
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \frac{xy}{x^2+y^2}$? $1$ $-1$ $0$ Limit does not exist

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.8k points)

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18

GATE2016-2-27
If $f(x)$ and $g(x)$ are two probability density functions, $f(x) = \begin{cases} \frac{x}{a}+1 & :-a \leq x < 0 \\ -\frac{x}{a}+1 & : 0 \leq x \leq a \\ 0 & :\text{otherwise} \end{cases}$ ... different; Variance of $f(x)$ and $g(x)$ are same Mean of $f(x)$ and $g(x)$ are different; Variance of $f(x)$ and $g(x)$ are different

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (11.8k points)

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19

GATE2016-2-28
The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.8k points)

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20

GATE2016-2-29
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.8k points)

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21

GATE2016-2-30
The quadratic approximation of $f(x)=x^3 – 3x^2 -5$ at the point $x=0$ is $3x^2 -6x-5$ $-3x^2-5$ $-3x^2+6x-5$ $3x^2-5$

asked Mar 28, 2018 in Numerical Methods by Milicevic3306 (11.8k points)

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22

GATE2017 CE-1: 1
The matrix $P$ is the inverse of a matrix $Q$. If $I$ denotes the identity matrix, which one of the following options is correct? $PQ=I$ but $QP \neq I$ $QP=I$ but $PQ \neq I$ $PQ=I$ and $QP= I$ $PQ-QP= I$

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (11.8k points)

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23

GATE2015-2-4
$\underset{x \to \infty}{\lim} \bigg( 1+ \frac{1}{x} \bigg)^{2x} $ is equal to $e^{-2}$ $e$ $1$ $e^2$

asked Mar 26, 2018 in Calculus by Milicevic3306 (11.8k points)

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24

GATE2015-1-26
The smallest and largest Eigen values of the following matrix are: $\begin{bmatrix} 3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5 \end{bmatrix}$ $1.5$ and $2.5$ $0.5$ and $2.5$ $1.0$ and $3.0$ $1.0$ and $2.0$

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (11.8k points)

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25

GATE2015-1-28
Consider the following differential equation: $x(y\:dx +x\:dy) \cos \frac{y}{x}=y(x\:dy-y\:dx) \sin \frac{y}{x}$ Which of the following is the solution of the above equation ($c$ is an arbitrary constant)? $\frac{x}{y} \cos \frac{y}{x} = c$ $\frac{x}{y} \sin \frac{y}{x} = c$ $xy \cos \frac{y}{x} = c$ $xy \sin \frac{y}{x} = c$

asked Mar 26, 2018 in Ordinary Differential Equation (ODE) by Milicevic3306 (11.8k points)

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1 answer

26

GATE2018 CE-1: 1
Which one of the following matrices is singular? $\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$ $\begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$ $\begin{bmatrix} 2 & 4\\ 3 & 6 \end{bmatrix}$ $\begin{bmatrix} 4 & 3\\ 6 & 2 \end{bmatrix}$

answered Feb 24, 2018 in Linear Algebra by goudarningu (360 points)

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1 answer

27

GATE2018 CE-2: 26
The matrix $\begin{pmatrix} 2 & -4 \\ 4 & -2 \end{pmatrix}$ has real eigenvalues and eigenvectors real eigenvalues but complex eigenvectors complex eigenvalues but real eigenvectors complex eigenvalues and eigenvectors

answered Feb 20, 2018 in Linear Algebra by paiyyawulpuja (120 points)

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1 answer

28

GATE2018 CE-1: 37
The solution at $x=1$, $t=1$ of the partial differential equation $\frac{\partial ^2 u}{\partial x^2} = 25 \frac{\partial ^2 u}{\partial t^2}$ subject to initial conditions of $u(0) = 3x$ and $\frac{\partial u}{\partial t}(0) =3$ is _______ $1$ $2$ $4$ $6$

answered Feb 19, 2018 in Partial Differential Equation (PDE): by praveenkumar_new (140 points)

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1 answer

29

GATE2018 CE-1: 38
The solution (up to three decimal places) at $x=1$ of the differential equation $\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + y =0$ subject to boundary conditions $y(0) = 1$ and $\frac{dy}{dx}(0) = -1$ is _____

answered Feb 18, 2018 in Ordinary Differential Equation (ODE) by pushpendratiwari1011 (1k points)

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1 answer

30

GATE2018 CE-2: 20
The quadratic equation $2x^2 - 3x +3=0$ is to be solved numerically starting with an initial guess as $x_0=2$. The new estimate of $x$ after the first iteration using Newton-Raphson method is ___

answered Feb 17, 2018 in Numerical Methods by Pushpendra Tiwari

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31

GATE2018 CE-2: 28
The rank of the following matrix is $\\ \begin{pmatrix} 1 & 1 & 0 & -2 \\ 2 & 0 & 2 & 2 \\ 4 & 1 & 3 & 1 \end{pmatrix}$ $1$ $2$ $3$ $4$

asked Feb 17, 2018 in Linear Algebra by gatecse (3.9k points)

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32

GATE2018 CE-2: 19
Probability (up to one decimal place) of consecutively picking $3$ red balls without replacement from a box containing $5$ red balls and $1$ white ball is _____

asked Feb 17, 2018 in Probability and Statistics by gatecse (3.9k points)

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33

GATE2018 CE-2: 1
The solution of the equation $x \frac{dy}{dx} +y = 0$ passing through the point $(1,1)$ is $x$ $x^2$ $x^{-1}$ $x^{-2}$

asked Feb 17, 2018 in Ordinary Differential Equation (ODE) by gatecse (3.9k points)

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34

GATE2018 CE-1: 26
The value of the integral $\int_0^{\pi} x \cos^2 x \: dx$ is $\pi^2/8$ $\pi^2/4$ $\pi^2/2$ $\pi^2$

asked Feb 17, 2018 in Calculus by gatecse (3.9k points)

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35

GATE2018 CE-1: 2
For the given orthogonal matrix Q, $Q = \begin{bmatrix} 3/7 & 2/7 & 6/7 \\ -6/7 & 3/7 & 2/7 \\ 2/7 & 6/7 & -3/7 \end{bmatrix}$ ... $\begin{bmatrix} -3/7 & 6/7 & -2/7 \\ -2/7 & -3/7 & -6/7 \\ -6/7 & -2/7 & 3/7 \end{bmatrix}$

asked Feb 17, 2018 in Linear Algebra by gatecse (3.9k points)

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36

GATE2018 CE-1: 3
At the point $x= 0$, the function $f(x) = x^3$ has local maximum local minimum both local maximum and minimum neither local maximum nor local minimum

asked Feb 17, 2018 in Calculus by gatecse (3.9k points)